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Chapter 4 Time Value of Money

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2 Future Value of Single Amount Deposit $1,000 into bank that pays 10% interest FV(1) = 1,000 + 1,000(.10) = 1,000(1.10) = 1,100 FV(2) = 1,000(1.10) + 1,000(1.10)(.10) = 1,000(1.10)(1.10) = 1,000(1.10) 2 = 1,210 FV(3) = 1,000(1.10) 2 + 1,000(1.10)2(.10) = 1,000(1.10) 2 (1.10) = 1,000(1.10) 3 = 1,331 FV(n) = PV(1+i) n

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3 Future Value of Single Amount After Year Accumulation 1 1,100 100 2 1,210 100 + 10 3 1,331 100 + 10 + 10 + 1 $5,000 deposited in bank paying 6% will earn after 8 years: FV = 5,000(1+.06) 8 = 5,000(1.59384807) = 7,969.24

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4 Present Value of Single Amount FV(n) = PV(1+i) n PV = FV/(1+i) n PV = FV(1+i) -n How much do you need to deposit today so you will have $10,000 accumulated after 7 years if you can earn 5% per year? PVS = 10,000(1+.05) -7 = 10,000(.710681233) = 7,106.81

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5 More Frequent Compounding The more often interest is added to your savings, the sooner you can start earning interest on your interest. Assume $1,000 invested at 10% per year: FV(1) = 1,000(1.10)1 = 1,100 [one year is one time period] FV(2) = 1,000(1+.10/2) 2 = 1,102.50 [one year is 2 six-month time periods: semi-annual] FV(4) = 1,000(1+.10/4) 4 = 1,103.81 [one year is 4 three-month time periods: quarterly compounding] FV(12) = 1,000(1+.10/12) 12 = 1,104.71 [one year is 12 one-month time periods: monthly compounding] FV(365) = 1,000(1+.10/365) 365 = 1,105.15 [one year is 365 daily time periods: daily compounding]

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6 More Frequent Compounding Effective Annual Rate EAR = (1 + i/m) m - 1 The accumulation for a single initial investment with frequent compounding is FVS = PV x (1+i/m) mn n = total number of years

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7 Future Value of an Annuity What a series of equal periodic cash flows will grow to 0 1 2 3 1,000 1,000 1,000 | | |___ 1,000(1.10) 0 = 1,000(1.00) = 1,000 | |________ 1,000(1.10) 1 = 1,000(1.10) = 1,100 |_____________ 1,000(1.10) 2 = 1,000(1.21) = 1,210 Total Accumulation = 3,310 FVA = A x (1+i) n - 1 = 1,000 x (1.10) 3 - 1 i.10 = 1,000x 3.31 = 3.310

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8 Present Value of an Annuity The starting amount that will create a series of equal periodic cash flows. 0 1 2 3 1,000 1,000 1,000 | | | 909.09 = 1,000(.90909) = 1,000(1.10)-1 ___| | | 826.44 = 1,000(.82644) = 1,000(1.10)-2 ________| | 751.31 = 1,000(.75131) = 1,000(1.10)-3 ______________| 2,486.84 = Starting amount needed PVA = A x 1 - (1+i) -n = 1,000 x 1 - (1.10) -3 i.10 = 1,000 x 2.48684 = 2,486.84

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PVA Example Deposit $2,486.84 in bank paying 10%: $2,486.84 (starting amount) 248.69 (first year interest) 2,735.53 (total end of first year) 1,000.00 (withdraw $1000) $1,000 1,735.53 (remaining amount 173.56 (interest second year) 1,909,09 (total end of second year) 1,000.00 (withdraw $1000) $1,000 909.09 (remaining amount) 90.91 (interest third year) 1,000.00 (total end of third year) 1,000.00 (withdraw $1000) $1,000 0 (remaining amount) 9

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10 Mortgages/Loans Mortgages and Loans are Present Value of an Annuity Problems PV = mortgage amount PMT = monthly payment N = number of monthly payments I/Y = annual interest rate P/Y = 12 C/Y = 12 FV = 0

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11 Computing Loan Payment Payment for $100,000 mortgage at 7% for 30 years. PVA = A x 1 - (1+i) -n i 100,000 = PMT x 1 - (1+.07/12) -360.07/12 100,000 = PMT / 150.3075679 PMT = 100,000/150.3075679 = 665.30

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12 Remaining Balance or Payoff The amount owed at any point in time is the present value of the remaining payments. The amount owed after five year would be: Balance = 665.30 x 1 - (1+.07/12) -300.07/12 100,000 = PMT x 141.4869034 Balance = 665.30 x 141.4869034 = 665.30

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BAII-Plus Calculator Solutions 13

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Basic Keys and Keystrokes Keys for doing time value of money and mortgage problems: STO~number: there are 10 memories (keys 1 through 9) RCL~number: recalls amount in memory +/ ‑ : key changes sign of value on screen (positive to negative) N: number of cash flows or payments I/Y: annual interest rate (in percent, e.g., 10 for 10%) PV: starting amount or loan amount PMT: periodic cash flow or loan payment FV: ending amount CPT: compute answer (solve for missing variable) 2nd~P/Y: number of payments per year 2nd~C/Y: number of compounding periods per year 2nd~BGN, 2 nd ~Set: changes cash flows to occur at end or beginning of each time period 14

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BAII-Plus Keystrokes (cont) BASIC KEYS USED IN FINANCE PROBLEMS When first starting calculations, you should reset calculator: ~ is used to separate key strokes (3~N: enter 3 then N key) 2 nd ~Reset~Enter (sets all values to default settings) The following two key sequences should be done before starting most new problem: 2nd~Quit: puts calculator in standard mode 2nd~CLR TVM: clears time value of money worksheet memories Some times you may wish to set to display two decimals: 2nd~Format~2~Enter: sets calculator to display two decimal places, though calculations in memory will be to thirteen places 15

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16 Future Value of Single Amount Calculator CALCULATOR: Future value of $1,000 after three years earning 10% per year Key Strokes: 2nd~Reset~Enter 3~N 10~I/Y 1000~PV CPT~FV [Answer: 1,331.00

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17 Present Value of Single Amount Calculator CALCULATOR: Present value, or amount you need to deposit today, so you will have accumulated $10,000 seven years from now earning 5% per year Key Strokes: 2nd~Rest~Enter 7~N 5~I/Y 10000~FV CPT~PV [Answer: 7,106.81]

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18 More Frequent Compounding Calculator CALCULATOR: Bank pays interest of 4%, compounded quarterly. If you deposit $2,000 and leave it in the bank for four years, how much will you have accumulated? Key Strokes: 2nd~Reset~Enter 2nd~P/Y~1~Enter ↓~4~Enter+CE/C 4~N 4~I/Y 2000~PV CPT~FV [Answer: 2,345.16]

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19 Future Value of an Annuity Calculator You deposit $2,000 each year for 10 years into a bank that pays 6% per year. How much will you have accumulated at the end of 10 years? Key Strokes: 2nd~Reset~Enter 2nd~P/Y~1~Enter (C/Y is changed to P/Y entry automatically) 10~N 6~I/Y 2000~PMT CPT~FV [Answer: 26,361.59]

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20 Present Value of an Annuity Calculator CALCULATOR: What amount do you need to deposit today so you an withdrawal $1,000 per year for three years while earning 6%. Key Strokes: 2nd~Reset~Enter 2nd~P/Y~1~Enter (C/Y is changed to P/Y entry automatically) 3~N 6~I/Y 1000~PMT CPT~PV [Answer: 2,673.01]

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21 More Frequent Compounding for an Annuity Calculator: What amount do you need to deposit today so you an withdrawal $1,000 per year for three years while earning 6%, compounded monthly. Key Strokes: for monthly compounding 2nd~Reset~Enter 2nd~P/Y~1~Enter ↓~12~Enter~C/CE 3~N 6~I/Y 1000~PMT CPT~FV [Answer: 2,664.74]

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22 Mortgage Problem CALCULATOR: Mortgage of $100,000 at 7% for 30 years, compute the monthly payment. 2nd~Reset~Enter~CE/C 2nd~I/Y, 12~Enter (sets P/Y and C/Y to 12) 360~N 7~I/Y 100,000~+/-~PV CPT~PMT [Ans: $665.30]

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23 Mortgage Amortization CALCULATOR: Amount owed after 5 years on 30 year mortgage of $100,000 at 7%. 2nd~Amort 49~Enter (P1 value: beginning year 5) ↓60~Enter (P2 value: end of year 5) ↓Bal = 94,131.76 (amount owed) ↓PRN = 1,342.91 (principle paid in year 5) ↓INT = 6,640.60 (interest paid in year 5)

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